Two-stroke engines of this kind are generally known. United States patent application publication 2009/0114172 A1, for example, discloses a two-stroke engine having a corresponding design. To achieve satisfactory cooling of the piston, the patent publication suggests arranging a flow-guide element such that it projects into the interior space of the crankcase and is oriented in the opposite direction to the rotation direction of the crankshaft. As a result, mixture from the crankcase is guided against the underside of the piston and in a direction toward the piston pin and thus cools the piston, the piston pin and the piston pin bearing.